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Thread: Irrational equation explanation

  1. #1
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    Irrational equation explanation

    I have a problem again with irrational equation.

    I have to solve this equation:
    $\displaystyle \sqrt[3]{x} + \sqrt[3]{{2x - 3}} = \sqrt[3]{{12(x - 1)}}$

    I have solve it like this:
    $\displaystyle \left( {\sqrt[3]{x} + \sqrt[3]{{2x - 3}}} \right)^3 = \left( {\sqrt[3]{{12(x - 1)}}} \right)^3$
    $\displaystyle x + 3\sqrt[3]{x}\sqrt[3]{{2x - 3}}(\sqrt[3]{x} + \sqrt[3]{{2x - 3}}) + 2x - 3 = 12x - 12 $
    $\displaystyle \sqrt[3]{x}\sqrt[3]{{2x - 3}}(\sqrt[3]{x} + \sqrt[3]{{2x - 3}}) = 3x - 3 $
    $\displaystyle \sqrt[3]{x}\sqrt[3]{{2x - 3}}\sqrt[3]{{12(x - 1)}} = 3x - 3 $
    $\displaystyle \sqrt[3]{{x(2x - 3)(12x - 12)}} = 3x - 3 $
    $\displaystyle x(2x - 3)4(3x - 3) = (3x - 3)^3 $
    $\displaystyle 4x(2x - 3) = (3x - 3)^2$
    $\displaystyle 8x^2 - 12x = 9x^2 - 18x + 9 $
    $\displaystyle x^2 - 6x + 9 = 0$
    $\displaystyle (x - 3)^2 = 0 $
    $\displaystyle x = 3$

    This is one solution but there is another one that is not obvious: $\displaystyle x=1$.

    Now, I have come to that solution by assuming that right side of equation is 0: $\displaystyle \sqrt[3]{{12(x - 1)}}=0$

    Can someone explain me how to know exactly what are ALL solutions of irrational equations?
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  2. #2
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    earboth's Avatar
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    Quote Originally Posted by DenMac21
    I have a problem again with irrational equation.

    I have to solve this equation:
    $\displaystyle \sqrt[3]{x} + \sqrt[3]{{2x - 3}} = \sqrt[3]{{12(x - 1)}}$

    I have solve it like this:
    [...snip...]

    $\displaystyle x(2x - 3)4(3x - 3) = (3x - 3)^3 $
    $\displaystyle 4x(2x - 3) = (3x - 3)^2$
    Hello,

    I've quoted those lines where you lost your 2nd solution.
    $\displaystyle x(2x - 3)4(3x - 3) - (3x - 3)^3 =0$
    $\displaystyle (3x-3) \cdot \left( 4x(2x - 3) - (3x - 3)^2 \right) =0$

    A product of two factors is zero, if one factor is zero:
    So you get:

    $\displaystyle (3x-3)=0 \ \vee \ 4x(2x - 3) - (3x - 3)^2 =0$

    After solving both equations you've got all possible results.

    Bye
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  3. #3
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    Quote Originally Posted by earboth
    Hello,

    I've quoted those lines where you lost your 2nd solution.
    $\displaystyle x(2x - 3)4(3x - 3) - (3x - 3)^3 =0$
    $\displaystyle (3x-3) \cdot \left( 4x(2x - 3) - (3x - 3)^2 \right) =0$

    A product of two factors is zero, if one factor is zero:
    So you get:

    $\displaystyle (3x-3)=0 \ \vee \ 4x(2x - 3) - (3x - 3)^2 =0$

    After solving both equations you've got all possible results.

    Bye
    Thanks
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